Asymptotic Solution of Moment Approximation of the Particle Population Balance Equation for Brownian Agglomeration

In the present study, the asymptotic solutions for particle moment approximation of population balance equation for Brownian agglomeration have been obtained analytically. At long time, the dimensionless particle moment (M_C) is an explicit monotonic decreasing function of fractal dimension in the free molecule regime, while it is a constant in the continuum regime. The asymptotic agglomerate growth rate is consistent with previous qualitative analysis and numerical solution, but the present asymptotic solution is more concise and straightforward.

Copyright 2015 American Association for Aerosol Research     

Asymptotic behavior of the effectiveness factor for variable activity catalysts

This paper studies the asymptotic behavior of the effectiveness factor for a catalyst with variable catalytic activity at large values of the Thiele modulus. The Liouvdle-Green (or WKBJ) approximation is employed to develop the asymptotic expansion of the solution of the linear diffusion-reaction equation. Asymptotic expressions are derived for variable catalytic activities in a catalyst pellet of a general geometry. These solutions are then used to obtain the asymptotic behavior of the effectiveness factor at large values of the Thiele modulus. The asymptotic results extend the concept of shape normalization to catalysts of nonuniform activity.     

Quasi-maximum likelihood estimation of periodic GARCH and periodic ARMA-GARCH processes

Abstract.  This article establishes the strong consistency and asymptotic normality (CAN) of the quasi-maximum likelihood estimator (QMLE) for generalized autoregressive conditionally heteroscedastic (GARCH) and autoregressive moving-average (ARMA)-GARCH processes with periodically time-varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution of the periodic GARCH (PGARCH) equation. As a result, it is shown that the moment of some positive order of the PGARCH solution is finite, under which we prove the strong consistency and asymptotic normality of the QMLE for a PGARCH process without any condition on its moments and for a periodic ARMA-GARCH (PARMA-PGARCH) under mild conditions.     

THE RESONANT SUPERCOMPRESSION AND SONOLUMINESCENCE OF A GAS BUBBLE IN A LIQUID-FILLED FLASK

The spherically-symmetric problem of the oscillations of a small gas bubble in the center of a spherical flask filled with a compressible liquid that is excited by small radial displacement of the flask's wall is considered. Two asymptotic solutions have been found for the low Mach number stage. The first one is an asymptotic solution for the field far from the bubble, and it corresponds to linear wave theory. The second one is an asymptotic solution for the boundary layer near the bubble and it corresponds to an incompressible fluid. In the analytical solution of the low Mach number step matching of these asymptotic solutions is done. A generalization of the Rayleigh = Plesset equation for a compressible liquid is given in the form of two ordinary difference-differential equations that take into account the pressure waves which are reflecting from the bubble and those that are incident on the bubble from the flask wall. The initial value problem for the initiation of the bubble oscillations due to flask wall excitation and for the evolution of these oscillations was considered. Linear and non-linear periodic bubble oscillations were analyzed analytically, and resonant frequencies were identified. Non-linear resonant and near-resonant solutions for the bubble's non-harmonic oscillations, which are excited by harmonic pressure or velocity oscillations on the flask wall, are obtained.     

The diffusion model of longitudinal mixing in beds of finite length. Numerical values H. Brenner, Chem. Engng Sci. 17: 229–243, 1962

Accurate numerical values are tabulated pertaining to the solution of the partial differential equation governing miscible fluid displacement dispersion phenomena in finite beds. Nondimensional results are furnished for the instantaneous exit solute concentration and average solute concentration remaining. Various asymptotic cases are discussed.     

幂律流体自然对流传质的研究

本文从解扩散微分方程组的角度研究非牛顿流体的传质问题,本构方程使用幂律模型,获得近似解,并利用电化学实验进行模拟和验证.在特定条件下,近似解可回复至牛顿流体的解.     

Recursive estimation: asymptotic confidence regions by empirical quantiles

Central limit theorems in stochastic approximation theory often involve an integral of a limit process of the observation errors. This integral is approximated by a further stochastic iteration. Then empirical quantiles are used to construct spherical asymptotic confidence regions for the parameter to be estimated. As an example the method is applied to a recursive solution of a linear equation in a separable real Banach space.     

CONVECTIVE TRANSPORT IN A ROTATING COILED TUBE

Regular perturbation techniques are used to study fully developed mass transfer in fluids flowing through rotating coiled tubes. A double series solution of the governing mass balance equation is developed for a constant wall flux boundary condition. The local, asymptotic mass transfer Nusselt number is computed from these results. Within the estimated region of validity of this solution the convective mass transfer coefficient is found to depend on essentially three independent dimensionless parameters. Mass transfer relative to the stationary coiled tube solution will either increase or decrease with rotation depending on the relative orientation of the fluid and rotational velocities.     

Deformation and breakup of a second-order fluid droplet in an electric field

The effects of elastic property on the deformation and breakup of an uncharged drop in a uniform electric field are investigated theoretically using the second-order fluid model as a constitutive equation. Two dimensionless numbers, the electric capillary number (C) and the Deborah number (De), the dimensionless parammeters governing the problem. The asymptotic analytic solution of the nonlinear free boundary problem is determined by utilizing the method of domain perturbation in the limit of small mathcal C and small De. The asymptotic solution provides the limiting point of C above which no steady-state drop shape exists. The linear stability theory shows that the elastic property of fluids give either stabilizing or destabilizing effect on the drop, depending on the deformation mode.     

Ice Nucleation Studies of Mineral Dust Particles with a New Continuous Flow Diffusion Chamber

The analytic representation of the exact coagulation kernel for combined Brownian motion and gravitational settling entails an infinite sum of ratios of modified Bessel functions. Heretofore, the evaluation of this sum has been limited to relatively small arguments due to computational difficulties, hindering the numerical solution of the general dynamic equation in computational aerosol transport simulations. An approximation and an asymptotic extension have been available, but their accuracy for large arguments has not been established, and they have been shown to lead to numerical diffusion and oscillation. Using multiple-precision arithmetic, exact values for the kernel are presented in this paper for large arguments, and it is shown that both the asymptotic and approximating formulae provide a poor fit in the range of practical interest. Based on the results, the asymptotic formula is rectified and it is shown that the approximation is no longer needed. Practical formulae are given to facilitate rapid numerical evaluation of the exact kernel for all arguments.     

Particle retardation in vertically oscillating fluids

An analytic expression has been deduced for the ultimate velocity of a free particle in a vertically oscillating fluid by evaluating asymptotic forms of Floquet's solution to the Mathieu equation arising from the nonlinear Langevin equation describing the particle-fluid interactions. Using general Mathieu theory it has been rigorously demonstrated that directional particle motion in a gravitational field can be retarded by sinusoidal fluid oscillations when the particle-fluid drag law is of square form. The earlier Rayleigh-Ritz approximation (Houghton 1966) for the retarded particle velocity is shown to be in good agreement with the results of Mathieu theory, as summarized in graphical form.     

An Exact Solution of the Fragmentation Equation

The accuracy of the self-similarity assumption employed in the study of the grinding equation is examined in detail. This is made possible by obtaining an exact solution for any homogeneous breakup function, thereby enabling the asymptotic limit as time proceeds to be examined carefully. For the Randolph-Ranjan model of breakup, we have obtained some explicit results and these have been employed to highlight the limitations of current self-similar solutions. In particular, we note that use of a self-similar solution which depends only on the zeroth and first moment of the distribution cannot give any detailed information on the higher moments. Nevertheless, at times very soon after the start of grinding, self-similarity does lead to useful and practical asymptotic results for size distributions, since it appears that higher moments are then of less importance. Thus the reason for the success of similarity is explained and the rate of approach to this condition is given. We have also introduced a new model of breakup which assumes that the minimum particle size in a given breakup process is always a fixed fraction of the initial size. This has the advantage of eliminating the divergence of the total number of particles produced per grinding action while still allowing the equation to be dealt with analytically. Finally, we discuss some steady state grinding distributions that arise from the new model.     

估量冷却水成垢的新模式

在大量循环冷却水模拟试验的基础上,对冷却水在换热器水侧所产生的污垢热阻进行了探讨.提出用一个可进行线性回归的、能反映诱导期的指数函数R_f=R_f~e-A/t-t.来描述污垢增长的规律.经将60轮模拟试验数据进行回归分析表明,这个指数函数能很好地与实验数据相符合.采用新的模式,就能由实验所得的R_f和相应的t值,直接回归出极限污垢热阻R_f~.     

A More Precise Criterion for the Formation of a Steady-State Sorption Front

Abstract

A more precise criterion for the formation of a steady-state sorption front is the convergence of the integrals which characterize the quantity of a sorbate in the leading edge of the front and the unused capacity of a sorbent behind the front. Existence of an asymptotic solution of the equation of sorption dynamics is not sufficient for convergence of these integrals, if the line passing through the points of the equilibrium isotherm, which correspond to the initial concentratons in the solution … and in the sorbent, is tangent to the isotherm at one of these points.     

SYMBOLIC OPERATOR SOLUTIONS OF LAPLACE'S AND STOKES’ EQUATIONS PART I LAPLACE'S EQUATION

Symbolic operator techniques are employed to derive a general solution of Laplace's equation in the infinite space external to a sphere. This is done for the case where the function vanishes on the sphere surface and arbitrary continuous asymptotic boundary data are imposed at infinity, such data being prescribed in the form of a solution of Laplace's equation that is analytic at the origin. In contrast with other standard methods for solving Laplace's equation, e.g., Green's functions, eigenfunction expansions, etc., the novelty of the proposed method lies in the fact that the solution can be expressed in a completely explicit form, directly in terms of (radial derivatives of)the given “undisturbed” field at infinity

A reciprocal theorem is derived and used to demonstrate that certain integral properties of the field can be obtained directly from the prescribed data at infinity, without recourse to a detailed solution of the relevant boundary-value problem. This global symbolic operator technique is illustrated for ellipsoidal as well as spherical particles

The elementary scalar harmonic analysis of the present paper serves as an entré to a companion paper (Part II), concerned with the application of similar symbolic techniques to the solution of more difficult vector biharmonic boundary-value problems, relevant to hydrodynamic Stokes flows in the infinite region external to a particle.     

FULLY DEVELOPED HEAT TRANSFER IN A ROTATING NON-ALIGNED STRAIGHT TUBE

Fully developed heat transfer in fluids flowing steadily through a rotating non-aligned straight tube is studied theoretically. A regular perturbation solution of the governing energy equation is developed for a constant wall flux boundary condition; this solution is valid only for “sufficiently mild” Coriolis disturbances. The asymptotic Nusselt number, complete to third order, is obtained from these results. Within the estimated region of validity of this solution the convective heat transfer coefficient depends on three independent dimensionless parameters. The first parameter serves as a Peclet number for energy transfer in the plane of the tube cross-section whereas the second and third parameters are proportional to the axial and transverse components of the Coriolis acceleration. Increasing the Coriolis acceleration will lead to transport enhancement relative to an analogous stationary straight tube exchange device.     

Condensation on a vertical rotating finned tube

Experiments have been carried out to determine the effect of centrifugal force on the rate of condensation of vapors. Methanol and isopropanol vapors were condensed on a vertical rotating finned tube and it was found that the data could be correlated by the equation: Condensation film heat transfer coefficients in excess of 5,700 W/(m² °K) and 4,200 W/(m² °K) were obtained for methanol and isopropanol respectively at a tube rotational speed of 1,000 rpm.     

Condensation on vertical porous metal fins

The problem of condensation of vapors on a porous plate fin is formulated and the dimensionless heat transfer coefficients are found to be dependent on two system variables, viz., the fin and the porosity parameters. Numerical simulations indicated that the condensation heat transfer coefficients for a porous fin are larger than those for a non-porous one of similar cross-section. For design purposes, a dimensionless equation for predicting the mean condensation heat transfer coefficient is proposed. The significant increase in the condensation heat transfer coefficient suggests that the condenser section of a flat plate heat pipe can be made more efficient in operation with the use of porous plate fins instead of impermeable fins of similar cross-section.     

Theoretical prediction of breakthrough curves for molecular sieve adsorption columns — II general isothermal solution for micropore diffusion control

Theoretical breakthrough curves for saturation and regeneration of a molecular sieve adsorption column are calculated from the solution of the appropriate diffusion equation for the case of micropore diffusion control. It is shown that with increasing non-linearity of the isotherm desorption becomes much slower than adsorption so that the regeneration cycle will generally be the limiting factor in column design. By comparing the breakthrough curves for saturation with the appropriate asymptotic solutions the conditions for validity of the ‘constant pattern’ approximation are obtained. Although restricted to lean gas mixtures containing only a single adsorbable component the analysis provides useful insight into column performance which should be of value in process design.